Neil Warnock's side laid siege to the Argyle goal for the majority of the match but could not find a way past inspired goalkeeper Romain Larrieu until Bromby broke the deadlock 12 minutes from time.
Gray bagged a second in the 85th minute as the Blades leapt up to sixth in the Coca-Cola Championship table despite David Friio's consolation.
The visitors made all the early running, and Ryan Dickson tested Paddy Kenny with a long-range effort which was comfortably dealt with by the Blades 'keeper before Friio, back in the Plymouth side after shaking off a virus, curled a free-kick narrowly wide.
It took 20 minutes for United to fashion a goalscoring opportunity when Andy Liddell's cross was headed straight at Larrieu by Danny Cadamarteri.
A minute later the hosts should have taken the lead, but Larrieu was well placed to keep out Gray's close-range header.
Phil Jagielka should have done better with a header from Liddell's corner, but the big defender failed to hit the target before Michael Tonge thumped an ambitious 30-yard effort wide of Larrieu's left-hand post.
Larrieu was also untroubled when Cadamarteri let fly from long range before Chris Morgan, back from suspension, rose above the Plymouth defence but somehow planted Tonge's pin-point free-kick wide.
Cadamarteri put yet another header wide from Liddell's cross as the first half drew to a close.
After the break United were straight back on the attack with Paul Shaw warming Larrieu's hands with a fierce strike before the former Gillingham man attempted an acrobatic overhead kick which was hacked clear by Mathias Doumbe.
Larrieu denied United again with a point-blank save from Cadamarteri, and moments later tipped a Shaw volley around the post.
Bromby was the next to try his luck, but the defender could not direct his header from Liddell's corner.
Cadamarteri appeared to be staging a one-man assault on the Argyle goal and he forced another save from Larrieu with a 20-yard curler before rattling the woodwork with a fierce strike as United's frustration grew.
With his goal by now under siege Larrieu pulled off another superb save, this time from Gray's header, before the hosts finally broke the deadlock.
Harley sent in a long throw from the left and Bromby headed his first goal for United to finally break the Frenchman's brave resistance.
Gray doubled the Blades' lead with a header from Liddell's corner before Friio set up an exciting finish, nodding home Lee Makel's cross to leave the hosts three minutes to hold on.